Proof involving a vacuously true statement

Proof involving a vacuously true statement

Let $S$ be a finite subset of a metric space. Show that it is closed.
I know a set is closed if and only if it contains all of its accumulation
points. Let $x$ be an accumulation point of $S$. I want to show that $x
\in S$. Since $S$ is a finite set, I know that it cannot have a
accumulation point. So does this imply that $x \in S$ and I am done with
the proof?
If I have made an error, could someone explain?